General Matrix-Valued Inhomogeneous Linear Stochastic Differential Equations and Applications

TL;DR

This paper derives solutions for general matrix-valued inhomogeneous linear stochastic differential equations and demonstrates their application in simplifying nonlinear stochastic equations to facilitate pathwise analysis.

Contribution

It generalizes previous scalar results to matrix-valued equations and introduces a method to reduce nonlinear equations to random differential equations.

Findings

Derived explicit solutions for matrix-valued inhomogeneous linear stochastic differential equations.

Reduced nonlinear stochastic equations to random differential equations for easier analysis.

Facilitated pathwise study of solutions through the reduction technique.

Abstract

The expressions of solutions for general $n\times m$ matrix-valued inhomogeneous linear stochastic differential equations are derived. This generalizes a result of Jaschke (2003) for scalar inhomogeneous linear stochastic differential equations. As an application, some $\R^n$ vector-valued inhomogeneous nonlinear stochastic differential equations are reduced to random differential equations, facilitating pathwise study of the solutions.